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Question

Suppose f(x)f(x)f′′(x)f(x)=0 where f(x) is continuously differentiable function with f(x)0 and satisfies f(0)=1 and f(0)=2, then f(x) is

A
x2+2x+1
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B
2ex1
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C
e2x
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D
4ex23
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Solution

The correct option is C e2x
Solving the given determinant, we get, f(x)2=f′′(x)f(x)
f′′(x)f(x)=f(x)f(x)
Integrating on both sides,
f(x)=k×f(x)
Since f(0)=1 and f(0)=2 k=2
f(x)f(x)=2
Now integrating on both sides,
ln(f(x))=2k×x
f(x)=e2kx
f(0)=2 k=1
So, option C is correct.

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